Hessian of a quadratic function
For quadratic functions, the Hessian (matrix of second-derivatives) is a constant matrix, that is, it does not depend on the variable 
.
As a specific example, consider the quadratic function
The Hessian is given by
![\begin{align*} \frac{\partial^2 q}{\partial x_i \partial x_j}(x) &= \left(\begin{array}{cc} \dfrac{\partial^2 q}{\partial x_1^2}(x) & \dfrac{\partial^2 q}{\partial x_1 \partial x_2}(x) \\[3ex] \dfrac{\partial^2 q}{\partial x_2 \partial x_1}(x) & \dfrac{\partial^2 q}{\partial x_2^2}(x) \end{array}\right) = 2\left(\begin{array}{ll} 8 & 3 \\ 3 & 4 \end{array}\right) \text{. } \end{align*}](https://pressbooks.pub/app/uploads/quicklatex/quicklatex.com-9839c8a77c1c4728c974ac7c212fd398_l3.png "Rendered by QuickLaTeX.com")
